As long as politics is not one-dimensional, there are completely reasonable cases where you get a rock-paper-scissors situation between 3 top candidates if voters select whomever is closest to them. That in turn violates the criteria that says the election result can't change if you add a non-winning candiate (rock beats scissors, but when paper enters the race now scissors is computed the winner).

This criteria is called "independence of irrelevant alternatives". A common criticism of Arrow's theorems usefulness is that it is a bit of a stretch to call paper an "irrelevant alternative" when rock-paper-scissors forms a cycle of preferences like that.

If politics is one-dimensional ("single-peaked preferences" in the article), then you never get this situation.

It's also important to note that a much more reasonable criteria exists called "local independence of irrelevant alternatives". This is the idea that total losers joining the race don't affect the result, however someone who is in the top rock-paper-scissors cycle (the "Smith Set") can still affect the result even if they don't win themselves. This is far more reasonable, as it's fairly arbitrary which of the candidates in that top cycle should win.

When there is no top cycle, the Smith Set is a single person (the "Condorcet Winner"). When there is a Smith Set, most reasonable voting systems will pick their winner somewhat arbitrarily as a member of the Smith Set, since everyone in the Smith Set beats everyone outside the Smith Set.

> Are the counterexamples offered by the theorem pathological, in the sense that they are unlikely to occur in practice but are theoretically possible? Or would they arise in practice frequently using standard rank voting systems?

Which particular problems occur, and the frequency with which they occur, depend on the particular voting system.

Plurality and majority/runoff (which are ranked preference voting systems with a vary narrow constraint on the preferences that are expressed on the input ballots, which is pretty much the same constraint as on the preferences reflected in the output of any single-winner voting system) hits problems fairly frequently in practice, but most of the common voting systems that most people think of as ranked-preference systems (IRV, etc.) still hit them in practice as well, though not generally as frequently and in ways which create as clear incentives to tactical voting.

The "worry" is not that this-or-that fantastical scenario might play out. It is that, as it turns out, the very mechanisms we use to measure group preferences simply cannot satisfy a list of very basic requirements. For example, the reason we do not use random drawings to determine who gets elected president is that we want the choice to "reflect" our preferences on the whole: but the result shows that this kind of "reflection" is probably not possible, and is always distorted in some fashion by the very procedures we adopt to make these decisions.

In "standard" FPTP-like systems we frequently see real-world cases that encourage tactical voting. E.g. look at the French Presidential election where Le Pen went through to the final round because there were four leftist candidates that split the leftist vote.